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" }], "Text", ImageRegion->{{0, 1}, {0, 1}}], Cell[CellGroupData[{Cell["\<\ DSolve[ {y'[x] == 2 y[x], y[0]==2}, y[x], x]\ \>", "Input", ImageRegion->{{0, 1}, {0, 1}}], Cell[OutputFormData["\<\ {{y[x] -> 2*E^(2*x)}}\ \>", "\<\ 2 x {{y[x] -> 2 E }}\ \>"], "Output", ImageRegion->{{0, 1}, {0, 1}}]}, Open]]}, Open]], Cell[CellGroupData[{Cell["Differential Equation in Two Variables", "Subsubsection", ImageRegion->{{0, 1}, {0, 1}}], Cell[CellGroupData[{Cell["\<\ DSolve[ {x'[t] == - y[t], y'[t] == x[t] }, {x[t],y[t]}, t] \ \>", "Input", ImageRegion->{{0, 1}, {0, 1}}], Cell[OutputFormData["\<\ {{x[t] -> (E^(-I*t)/2 + E^(I*t)/2)*C[1] + (-I/2*E^(-I*t) + I/2*E^(I*t))*C[2], y[t] -> (I/2*E^(-I*t) - I/2*E^(I*t))*C[1] + (E^(-I*t)/2 + E^(I*t)/2)*C[2]}}\ \>", "\<\ -I t I t E E {{x[t] -> (----- + ----) C[1] + 2 2 -I -I t I I t (-- E + - E ) C[2], 2 2 -I t I t I -I t I I t E E y[t] -> (- E - - E ) C[1] + (----- + ----) C[2]} 2 2 2 2 }\ \>"], "Output", ImageRegion->{{0, 1}, {0, 1}}]}, Open]]}, Open]], Cell[CellGroupData[{Cell["Second Order Differential Equation", "Subsubsection", ImageRegion->{{0, 1}, {0, 1}}], Cell[CellGroupData[{Cell["DSolve[y''[x] - k y[x] == 1, y[x], x]", "Input", ImageRegion->{{0, 1}, {0, 1}}], Cell[OutputFormData["\<\ {{y[x] -> -k^(-1) + C[1]/E^(k^(1/2)*x) + E^(k^(1/2)*x)*C[2]}}\ \>", "\<\ 1 C[1] Sqrt[k] x {{y[x] -> -(-) + ---------- + E C[2]}} k Sqrt[k] x E\ \>"], "Output", ImageRegion->{{0, 1}, {0, 1}}]}, Open]]}, Open]], Cell[CellGroupData[{Cell["A Note on Pure Function Solutions", "Subsubsection", ImageRegion->{{0, 1}, {0, 1}}], Cell[TextData[{ "When you ask ", StyleBox["DSolve", FontFamily->"Courier", FontSize->12, FontWeight->"Bold"], " to get a solution for ", StyleBox["y[x]", FontFamily->"Courier", FontSize->12, FontWeight->"Bold"], ", the rules it returns specify how to replace ", StyleBox["y[x]", FontFamily->"Courier", FontSize->12, FontWeight->"Bold"], " in any expression. If you want to manipulate solutions that you get from \ ", StyleBox["DSolve", FontFamily->"Courier", FontSize->12, FontWeight->"Bold"], ", you will often find it better to ask for solutions for ", StyleBox["y", FontFamily->"Courier", FontSize->12, FontWeight->"Bold"], ", rather than ", StyleBox["y[x]", FontFamily->"Courier", FontSize->12, FontWeight->"Bold"], ". In this case, the solution is returned as a \"pure function\". This is \ illustrated in the next example. For more information on how the ", StyleBox["Function", FontFamily->"Courier", FontSize->12, FontWeight->"Bold"], " object returned by ", StyleBox["DSolve", FontFamily->"Courier", FontSize->12, FontWeight->"Bold"], " works, see page 207." }], "Text", ImageRegion->{{0, 1}, {0, 1}}]}, Open]], Cell[CellGroupData[{Cell["Pure Function Solution to a Differential Equation", "Subsubsection", ImageRegion->{{0, 1}, {0, 1}}], Cell[TextData[{ "This computes the solution of the differential equation,", StyleBox[" y'[x] == 2 y[x]", FontFamily->"Courier", FontSize->12, FontWeight->"Bold"], " and returns the solution as a pure function. In the next line, the \ solution is evaluated at x = 2.8. The last input substitutes the solution \ into ", StyleBox["y'[x] - 2 y[x]", FontFamily->"Courier", FontSize->12, FontWeight->"Bold"], ". The result of 0 confirms that we have found the solution." }], "Text", ImageRegion->{{0, 1}, {0, 1}}], Cell[CellGroupData[{Cell["soln = DSolve[ {y'[x] == 2 y[x], y[0] == 1}, y, x]", "Input", ImageRegion->{{0, 1}, {0, 1}}], Cell[OutputFormData["\<\ {{y -> Function[x, E^(2*x)]}}\ \>", "\<\ 2 x {{y -> Function[x, E ]}}\ \>"], "Output", ImageRegion->{{0, 1}, {0, 1}}]}, Open]], Cell[CellGroupData[{Cell["y[2.8] /. soln", "Input", ImageRegion->{{0, 1}, {0, 1}}], Cell[OutputFormData["\<\ {270.4264074261526}\ \>", "\<\ {270.426}\ \>"], "Output", ImageRegion->{{0, 1}, {0, 1}}]}, Open]], Cell[CellGroupData[{Cell["y'[x] - 2 y[x] /. soln", "Input", ImageRegion->{{0, 1}, {0, 1}}], Cell[OutputFormData["\<\ \ \>", "\<\ {0}\ \>"], "Output", ImageRegion->{{0, 1}, {0, 1}}]}, Open]]}, Open]], Cell[CellGroupData[{Cell["A Note on Numerical Solutions to Differential Equations", \ "Subsubsection", ImageRegion->{{0, 1}, {0, 1}}], Cell[TextData[{ "Most differential equations do not have a \"closed form\" solution. In \ these cases you can use ", StyleBox["NDSolve", FontFamily->"Courier", FontSize->12, FontWeight->"Bold"], " to generate a numerical approximation of the solution. ", StyleBox["NDSolve", FontFamily->"Courier", FontSize->12, FontWeight->"Bold"], " represents solutions as ", StyleBox["InterpolatingFunction", FontFamily->"Courier", FontSize->12, FontWeight->"Bold"], " objects. The ", StyleBox["InterpolatingFunction", FontFamily->"Courier", FontSize->12, FontWeight->"Bold"], " objects provide approximations to the solutions over the range of the \ independent variable. For more information on ", StyleBox["InterpolatingFunction", FontFamily->"Courier", FontSize->12, FontWeight->"Bold"], " objects, see page 676." }], "Text", ImageRegion->{{0, 1}, {0, 1}}]}, Open]], Cell[CellGroupData[{Cell["Numerical Solution of a Differential Equation", "Subsubsection", ImageRegion->{{0, 1}, {0, 1}}], Cell[TextData[{ "This finds a numerical solution to the differential equation \ ", StyleBox["y''[x] + Sin[x]^2 y'[x] + y[x] == Cos[x]^2 ", FontFamily->"Courier", FontSize->12, FontWeight->"Bold"], "with initial conditions ", StyleBox["y[0] == 1 and y'[0] == 0 ", FontFamily->"Courier", FontSize->12, FontWeight->"Bold"], "and x in the range of 0 to 10. 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